Uniform Law on the Unit Sphere of a Banach Space

Transcription

1 Uniform Law on the Unit Shere of a Banach Sace by Bernard Beauzamy Société de Calcul Mathématique SA Faubourg Saint Honoré Paris France Setember 008 Abstract We investigate the construction of a uniform robability law on the unit shere of a finite-dimensional, real, rearrangement-invariant Banach sace We show that this construction is ossible under a geometrical condition, and it comes from a robability on : there is a density f, on, such that if,, are indeendent variables following this law, the law of equied with this norm,,,, is the uniform law on the unit shere of, I Introduction For ractical uroses, one may want to generate a samle of oints on the unit shere of a Banach sace Here are two eamles : Choosing a direction at random is useful for the search of solutions of PDE s : one starts at a given oint and then moves to some direction In this case, the norm is usually the Euclidean norm See the acknowledgements at the end of the aer for our original motivation, in the framework of two contracts with IRSN (France) Choosing a roortion at random is useful, for instance in economics : a budget may deend on some goods, and one wants to study the variations of the budget, deending on various weights on the goods In this case, the norm is usually the sum of absolute values of the coefficients, that is the l norm In all cases, one deals with a finite-dimensional real Banach sace, that is the sace equied with some norm In general, one does not have any redefined wish or reference about the oints which must be chosen : they need to be on the unit shere, but no direction is rivileged This means that the law we want for our samle is the uniform law on the unit shere Bernard Beauzamy : Uniform law on the unit shere, July 008

2 Our final observation, about the motives, is that usually one wants to use a law on the real numbers (building a law directly on the unit shere of a Banach sace is very comlicated) : one wants some density function f, on the real numbers, such that if we draw a samle,, using this law, and then normalize it correctly, that is relace it by,,,, law, then the corresonding oints on the unit shere will follow a uniform II A reliminary negative remark First, we observe that such a law cannot be obtained from a uniform law on each comonent searately Indeed, take the Euclidean norm First draw a samle for a random vector (,, ) ; each k following a uniform law, for instance on the interval 0,, and then normalize, that is relace each k by : y k k i i, so that yk k The vector y,, y is indeed on the unit shere of l, but does not follow a uniform law Let us see this on a very simle eamle, in dimension We take two random numbers and with uniform law between 0 and This means that we have a uniform law in the unit square 0, For each and, we consider the normalized vector : y, y The law of this vector has density : If 0, f ( ) 4 cos ( ) If, f ( ) 4 sin ( ) This is quite different from a uniform law This remark is quite general : a uniform law in the square does not roject to a uniform law on the unit shere, no matter what the norm is Bernard Beauzamy : Uniform law on the unit shere, July 008

3 III Uniform law on a shere What we mean here by uniform law means uniform with resect to Lebesgue measure It means that the robability to fall in some set deends only on the Lebesgue measure of this set, and not on the osition on the set on the unit shere Since Lebesgue measure is invariant under ermutation of the coordinates, we investigate only norms which have the same roerty More recisely, we assume that our Banach sace is equied with a norm which satisfies : (),, ( ),,, for any ermutation of the set,, This is also necessary since we want our law to derive from a law on the real numbers : we draw a samle and then normalize This imlies that all coordinates have the same law Let h(,, ) be the density of robability we want to build Then, the fact that we have a uniform law in a rearrangement-invariant sace means that this function h will be constant on its domain of definition : h(,, ) does not deend on (,, ), rovided (,, ) This is the fundamental roerty we will use We now show how to build a uniform distribution on the unit shere, starting from a robability density on the real line We observe that it is enough to construct our robability density for ositive variables Indeed, there are sets of equal measure, deending on the sign of each k, and the construction of the density in one of them transfers to all the others The general construction we give is insired (with corrections) by the one given in the book by Luc Devroye [], chater 5, theorem, for the l norm Let : be the ositive art of the unit shere C,, ; 0 k,,, k We need to introduce a restriction for the norm, which is of geometrical nature In order to elicit this condition, we investigate the condition : for given s ;,,, that is as a condition on alone,,, s () First, we observe that () may never be satisfied, if s is too small or,, too large Let : Bernard Beauzamy : Uniform law on the unit shere, July 008 3

4 be the set on which () can be satisfied A ( s),, ;,,, s If there is a oint for which () holds, then there is a unique with We denote this oint by s s,,, ;,, In other words, ;,, ma ;,,, s s and we have s ;,, if 0 satisfies : For instance, for the Euclidean norm, and,,, s s ;,, s / A ( s),, ; s is the ball of radius s in the Euclidean dimensional sace We can now state our result : Theorem - Let be a norm on, invariant under ermutation of the variables Then, there is a robability density on such that, if,, are indeendent variables following this law, the law of is the uniform law on the unit shere,,,, of, equied with this norm, if and only if the values of s; z values of s; z function G such that : and of deend only on the In other words, the answer is ositive if and only if there is a s ; z G, s ; z () Bernard Beauzamy : Uniform law on the unit shere, July 008 4

5 Proof of the Theorem We set : S,, This is a ositive random variable We also define :, Z,, z,, f z s be the density of the joint law of the ule, Let ZS, (, ) comuted as a conditional robability, knowing Z, that is : Z, S S Z Z ZS This density can be f ( z, s) f ( s) f ( z) (3) where fs Z() s denotes the conditional robability density of S knowing Z and fz () z is the density of Z The roof of the Theorem will be divided into three stes Ste Under the above assumtion, the density fzs, ( z, s ) is indeendent of z (,, ) So we are looking for an identity of the following form : where the right-hand side is indeendent of z (,, ) We have, since the variables,, are indeendent : fs Z ( s) fz ( z) c( s) (4) fz( z) f ( ) f ( ) and that is : F ( s) P,, s,, S Z k F ( s) P,,, s S Z This robability is entirely determined by the law on the values of,, are fied, since the norm is known and Bernard Beauzamy : Uniform law on the unit shere, July 008 5

6 The condition : is equivalent to the fact that s,,, is in the interval : 0 s ;,,, by definition of the function, rovided that,, A ( s) For instance, for the Euclidean norm, the condition is equivalent to : rovided that : s,,, / 0 s s Let f be the density of robability we are looking for The identity (4) is equivalent to : is indeendent of (,, ) f s ;,, f ( ) f ( ) c( s) (5), rovided that,, A ( s) Since all variables are equivalent, it is enough to ensure this for Comuting the derivative with resect to this variable, we get : that is, f s ;,, s ;,, f ( ) f ( ) f s ;,, f ( ) f ( ) f ( ) 0 f s ;,, s ;,, f ( ) f s ;,, f ( ) (6) This can be written : s ; z s; z f s ; z f( ) f f( ) We are looking for a density function f satisfying equation (7) Set : Bernard Beauzamy : Uniform law on the unit shere, July 008 (7) 6

7 g ( ) f( ) f( ) This is the logarithmic derivative of the function f From the identity (7), we deduce : g s ; z g( ) s; z (8) In order to understand what this equation means, let us come back for a while to the case of the Euclidean norm In this case, we have : s and the identity (8) becomes : s g s g( ) (9) / Taking, 0, we get, for s : / g s c s for some constant c This means, for all u 0 : which gives g u cu Log f c u and : u f ( u) ec Now, the value of c is determined by normalization : the integral of f must be, and we find : u f( u) e 4, for u 0 Bernard Beauzamy : Uniform law on the unit shere, July 008 7

8 The same argument holds in general The identity (8) allows the construction of the function g : fi any value for, for instance Let c be the unknown value of g ( ) Fi u 0 and find s,,, such that ( s ;,,, ) u (this is always ossible) : then the identity (8) defines gu ( ) From the definition of g, one deduces f, u to a multilicative constant, which is comuted by normalization (the integral of f must be ) This concludes our construction However, there is a comatibility condition ; the identity : g s ; z g( ) s; z must give the same value to g when the value of ( sz, ) is fied This means that the value of the derivative s; z, for fied, must deend only on the value of s; z In other words, there must eist a function G such that : s ; z G, s ; z (0) This is the case for the Euclidean norm, since : s; z s ; z and the same holds for all the l norms, We note that condition (0) does not need to be satisfied for any norm on the sace Indeed, a norm on is defined by a symmetric conve body in this sace (see for instance B Beauzamy []), and condition (0) means that the value of the derivative s; z, for fied, deends only on the value of s; z Let us see what this means geometrically, in 3 dimensions : Bernard Beauzamy : Uniform law on the unit shere, July 008 8

9 D C B A Figure : sloes of the tangent to the unit shere at two different oints On this icture, the two oints A and B have the same value for the first coordinate, which is the length AC BD But the tangent to the and the same value for ;, unit shere does not need to be have the same sloe in C and in D This concludes Ste of the roof Ste We just showed that fzs, ( z, s ) was indeendent of z (,, ) A () s Let U,, S S indeendent of u, in the set () f u s be the joint density of, and let, (, ) US A, since : f ( u, s) f ( us, s), U, S Z, S, for z in the set US ; this density is for all s 0 The density of U itself can be obtained, integrating with resect to s that is : f ( u) f ( u, s) ds () U 0 U, S and we see that this value is constant on the whole set A () Bernard Beauzamy : Uniform law on the unit shere, July 008 9

10 But now, since the density of,, is constant, so is the density of S S,,, ;,,,,, S S S S S S S () This concludes the roof of the Theorem Corollaries We now show how this general result alies to secific norms : a) The case of the Euclidean norm This case has been treated along with the roof of the Theorem We obtain : Theorem -Let and let,, be indeendent normal variables (mean 0, variance ) Then the vector :,, follows a uniform law on the unit shere of the dimensional Euclidean sace In other words, in order to obtain a uniform law on the shere, one should not start with uniform variables, but with Gaussian A roof of this result can be found in the book by R J Muirhead [3], 37 (communicated by Paul Deheuvels) This roof uses the fact that the normal law on is inva- riant under all orthogonal transformations of the sace into itself So, its rojection uon the unit shere is itself invariant under the rojections of the transformations The unit shere is a locally comact grou for these transformations, and a general result says that there is only one invariant Haar measure, which must be the uniform law b) The case of the l norm This is treated in the book by Luc Devroye [] ; we get : Theorem 3 -Let and let,, be indeendent variables following an eonential law Then the vector :,, follows a uniform law on the ositive art of the unit shere of the dimensional l sace Bernard Beauzamy : Uniform law on the unit shere, July 008 0

11 c) The case of the l norms, This is a generalization of the revious ones The case is obvious (this is just the uniform law in the unit ball) For the case, we get : Theorem 4 -Let and let,, be indeendent variables following a law with density e t t 0 Then the vector :,, / / follows a uniform law on the ositive art of the unit shere of the dimensional l sace Proof of Theorem 4 We indicate briefly how this density is obtained In this case, we have : s ;,, s / We have : so condition (0) is satisfied The identity (8) becomes : / / s g s g( ) (3) Taking, 0, we get, for s : / g s c s for some constant c This means, for all u 0 : which gives cu g u Log f u c Bernard Beauzamy : Uniform law on the unit shere, July 008

12 and : u f ( u) ec The value of c should be determined by normalization ; of course c 0, but we may just as well take : f ( u) e u, for u 0, since we divide at the end by the quantity / Theorem 4 This concludes the roof of References [] B Beauzamy : Introduction to Banach Saces and their Geometry North Holland, Collection «Notas de Matematica», vol 68, second edition,985 [] Luc Devroye : Non Uniform Random Variate Generation, Sringer-Verlag, New York, 986 [3] R J Muirhead : Asects of Multivariate Statistical Theory Wiley, New York, 98 Acknowledgements We thank Paul Deheuvels for indicating the book by Luc Devroye, and Luc Devroye for his comments about the roofs Originally, our reoccuation about "random directions" in a Banach saces comes from two contracts we had with the "Institut de Radiorotection et de Sûreté Nucléaire" (IRSN, France) : R50/0609 (7//006) and DA 7960/CA (/03/008) ; they deal with the safety of nuclear reactors, and the question was : in a comutational code deending uon many arameters, find the configuration which might lead to the highest temerature The main tool was the "Eerimental Probabilistic Hyersurface", which rovides a robability law for the temerature, at each oint of the configuration sace See htt://wwwscmsacom/rmm/rmm_ephhtm for more details Bernard Beauzamy : Uniform law on the unit shere, July 008